-2x^2+3x=4x-15 — step-by-step solution

1) Let's solve in

ℝ

- 2 x^{2} + 3 x = 4 x - 15

- 2 x^{2}

+ 3 x

- 4 x

+ 15 = 0

ⓘ Like terms are grouped by adding their coefficients:

3 x - 4 x

- x

- 2 x^{2} - x + 15 = 0

Since

-1 \neq 0

, divide both sides by

-1

2 x^{2} + x - 15 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = 1

and

c = -15

Δ = 1 - 4 \cdot 2 \cdot (-15)

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Two minus signs cancel out, the product of two negative numbers is positive:

- 8 \cdot (-15)

8 \cdot 15

Δ = 1 + 8 \cdot 15

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 15

8 \cdot (10 + 5)

8 \cdot 10 + 8 \cdot 5

80 + 40

120

Δ = 1 + 120

ⓘ Start from

1

and move forward

120

on the number line:

1 + 120

121

Δ = 121

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 1 - \sqrt{121}}{2 \cdot 2}

x_{1} = \frac{- 1 - \sqrt{121}}{2 \cdot 2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{121}

11

;

11^{2}

121

ⅰ

2 \cdot 2

4

x_{1} = \frac{- 1 - 11}{4}

ⓘ The numerator and the denominator are computed before the division:

- 1 - 11

-12

x_{1} = - \frac{12}{4}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

4

, their greatest common divisor:

- \frac{12}{4}

-3

x_{1} = -3

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 1 + \sqrt{121}}{2 \cdot 2}

x_{2} = \frac{- 1 + \sqrt{121}}{2 \cdot 2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{121}

11

;

11^{2}

121

ⅰ

2 \cdot 2

4

x_{2} = \frac{- 1 + 11}{4}

ⓘ The numerator and the denominator are computed before the division:

- 1 + 11

10

x_{2} = \frac{10}{4}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{10}{4}

\frac{5 \cdot 2}{2 \cdot 2}

\frac{5}{2}

x_{2} = \frac{5}{2}

▶Solutions obtained:

x = {-3, \frac{5}{2}}

2) The numeric value of:

x = {-3, \frac{5}{2}}

equals:

▷

x = {-3, 2.5}